Spiral polynomial division multiplexing

ABSTRACT

A method for communicating using polynomial-based signals. In such a method, a set of basis polynomial functions used to generate waveforms may be identified, wherein each of the basis polynomial functions in the set of basis polynomial functions is orthogonal to each of the other basis polynomial functions in the set of basis polynomial functions in a coordinate space. The set of basis polynomial functions may be combined into a message polynomial. The message polynomial may be convolved with a reference polynomial to produce a transmission polynomial. From the transmission polynomial, a sequence of amplitude values may be generated. Finally, a signal may be transmitted based on the sequence of amplitude values, which may be further modified based on, for example, instantaneous spectral analysis. In some embodiments, orthogonal polynomials may include Chebyshev or Cairns polynomials.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority from U.S. patent application Ser. No.15/255,944, filed on Sep. 2, 2016, entitled “Spiral Polynomial DivisionMultiplexing”, U.S. Provisional Patent Application No. 62/213,418, filedon Sep. 2, 2015, entitled “Spiral Polynomial Division Multiplexing,” andfrom U.S. Provisional Patent Application No. 62/256,532, filed on Nov.17, 2015, entitled “Method for Determining Instantaneous SpectralUsage,” the entire contents of which are hereby incorporated byreference.

BACKGROUND

Applicant's prior U.S. Pat. No. 8,472,534 entitled “TelecommunicationSignaling Using Non-Linear Functions” and U.S. Pat. No. 8,861,327entitled “Methods and Systems for Communicating”, the contents of whichare herein incorporated by reference in their entirety, introducedspiral-based signal modulation. Spiral-based signal modulation may basesignal modulation on complex spirals, rather than the traditionalcomplex circles used by standard signal modulation techniques such asQuadrature Amplitude Modulation (QAM) and Phase-Shift Keying (PSK).

Multiplexing refers to the combination of multiple digital data streamsor sub-channels into a single signal for communication over a sharedchannel. In the context of wireless communication, multiplexing providesa means to share an expensive resource, spectrum. Additionally, breakinga channel into sub-channels may facilitate resistance to channelimpairments such as fading.

Any multiplexing technique requires a method for the multiplexer (MUX)in the transmitter to combine sub-channels into a single signal, and forthe demultiplexer (DMX) in the receiver to reverse the process to obtainthe original data streams.

Existing well-known and widely-deployed multiplexing methods includeCode-Division Multiplexing (CDM), Orthogonal Frequency-DivisionMultiplexing (OFDM), and Time-Division Multiplexing (TDM). However, inall cases, these underlying sub-channel signal modulation techniques arebased on sinusoidals. The overall framework is that sub-channelsinusoidal modulations are combined into a higher-level sinusoidalmodulation in the common channel.

SUMMARY

According to an exemplary embodiment, a modulation scheme that ispolynomial-based rather than sinusoidal-based may be disclosed.

Applicant's prior patents noted above disclosed an exemplaryimplementation of spiral modulation in which spiral-based symbolwaveforms were formed through multiplying a complex circle by a risingexponential (the “head function”) for part of the symbol time,optionally connected to a “tail function” that returned the amplitude toits original value. The intra-symbol amplitude variation was thereforedefined, at least for the “head function” portion of the symbol time, bythe properties of an exponential.

According to an exemplary embodiment, a method for communicating may bedisclosed. Such a method may include: identifying a set of basispolynomial functions used to generate waveforms, wherein each of thebasis polynomial functions in the set of basis polynomial functions isorthogonal to each of the other basis polynomial functions in the set ofbasis polynomial functions in a coordinate space; combining the set ofbasis polynomial functions into a message polynomial; convolving themessage polynomial with a reference polynomial to produce a transmissionpolynomial; generating, from the transmission polynomial, a sequence ofamplitude values; and transmitting, with a transmitter, a signal basedon the sequence of amplitude values. In some embodiments, orthogonalpolynomials may include Chebyshev or Cairns polynomials.

In an embodiment, the method may further include converting, usinginstantaneous spectral analysis, a sequence of amplitude values into aset of sinusoidals with continuously-varying amplitude; and generatingthe signal based on the sequence of amplitude values by combining theset of sinusoidals with continuously-varying amplitude. In anembodiment, the step of converting using Instantaneous Spectral Analysis(ISA) may entail projecting a transmission polynomial onto the Cairnsseries functions; converting the polynomial from a function described bythe Cairns series functions to a function described by the Cairnsexponential functions; and combining, into a sum of sinusoidals withcontinuously time-varying amplitudes, frequency information containedwithin the function described by the Cairns exponential functions. (Inanother embodiment, for example an embodiment in which the transmissionpolynomial is not known or not accessible to a device implementing ISA,the step of converting using ISA may further include fitting atransmission polynomial to a sequence of amplitude values.)

In an embodiment, the method may further include polynomial-basedsynchronization using full-power synchronization, limited-powersynchronization, or no-power synchronization, where the power level mayrefer to the amount of available signal power devoted exclusively tosynchronization over some period of time. The method may further includeperforming dimension reduction to mitigate coherent interferencerejection or to reduce Peak-to-Average Power Ratio (PAPR).

In another embodiment, a communication, such as might be sent via thefirst method, may be received and decoded using another method. Such amethod may include receiving a signal with a receiver, the signal havinga plurality of sub-channels and including a sequence of amplitudevalues; and demultiplexing the signal. The step of demultiplexing thesignal may include fitting a reconstructed transmission polynomial tothe sequence of amplitude values; deconvolving, from the reconstructedtransmission polynomial, a reference polynomial, yielding a transmittedmessage polynomial; determining, by orthogonal projection of thetransmitted message polynomial into a polynomial coefficient space, aplurality of sub-channel amplitudes of the signal; generating a combinedbit sequence of the signal by mapping the plurality of sub-channelamplitudes into bit sequences and combining the bit sequences; andoutputting the combined bit sequence.

In another exemplary embodiment, each of the method for sending and themessage for receiving communications may be embodied on a particularsystem or particular apparatus. Such a system may include a spiralpolynomial division multiplexer and a transmitter, the spiral polynomialdivision multiplexer being configured to: identify a set of basispolynomial functions used to generate waveforms, wherein each of thebasis polynomial functions in the set of basis polynomial functions isorthogonal to each of the other basis polynomial functions in the set ofbasis polynomial functions in a coordinate space; combine the set ofbasis polynomial functions into a message polynomial; convolve themessage polynomial with a reference polynomial to produce a transmissionpolynomial; and generate, from the transmission polynomial, a sequenceof amplitude values. The transmitter may be configured to transmit asignal based on the sequence of amplitude values.

BRIEF DESCRIPTION OF THE FIGURES

Advantages of embodiments of the present invention will be apparent fromthe following detailed description of the exemplary embodiments thereof,which description should be considered in conjunction with theaccompanying drawings in which like numerals

FIG. 1 may show an exemplary plot of a Cairns function having a risingexponential.

FIG. 2 may show an exemplary plot of a Cairns function having a fallingexponential.

FIG. 3 may show an exemplary plot of a Cairns series function.

FIG. 4 may show an exemplary plot of a Cairns series function.

FIG. 5 may show an exemplary plot of a Cairns series function.

FIG. 6 may show an exemplary plot of a Cairns series function.

FIG. 7 may show an exemplary plot of a Cairns series function.

FIG. 8 may show an exemplary plot of a Cairns series function.

FIG. 9 may show an exemplary plot of a transmission polynomial.

FIG. 10 may show an exemplary plot of a transmission polynomial.

FIG. 11 may show an exemplary plot of a reference polynomial.

FIG. 12 may show an exemplary plot of a synchronization phase match.

FIG. 13 may show an exemplary plot of a synchronization phase mismatch.

FIG. 14 may show an exemplary plot of a synchronization phase mismatch.

FIG. 15 may show an exemplary plot of the Euclidean distances ofpossible synchronization phases.

FIG. 16 may show an exemplary plot of a synchronization phase match.

FIG. 17 may show an exemplary plot of a synchronization phase mismatch.

FIG. 18 may show an exemplary plot of a synchronization phase mismatch.

FIG. 19 may show an exemplary plot of the Euclidean distances ofpossible synchronization phases.

FIG. 20 may show an exemplary plot of a synchronization phase match.

FIG. 21 may show an exemplary plot of a synchronization phase mismatch.

FIG. 22 may show an exemplary plot of a synchronization phase mismatch.

FIG. 23 may show an exemplary plot of the Euclidean distances ofpossible synchronization phases.

FIG. 24 may show an exemplary plot of the values at which Taylor termsfirst produce larger values than the Taylor terms of next lower degree(the “Taylor term amplitude cross-overs”).

FIG. 25 may show exemplary MATLAB code for the generation of a randompolynomial.

FIG. 26 may show exemplary MATLAB code for the conversion of apolynomial to a Taylor polynomial.

FIG. 27 may show exemplary MATLAB code for zero-padding a Taylorpolynomial.

FIG. 28 may show exemplary MATLAB code for generating normalizationcoefficients.

FIG. 29 may show exemplary MATLAB code for generating a Cairnsprojection matrix.

FIG. 30 may show exemplary MATLAB code for projecting a Taylorpolynomial onto Cairns space.

FIG. 31 may show exemplary MATLAB code for determining amplitude values.

FIG. 32 may show exemplary MATLAB code for finding a row index.

FIG. 33 may show exemplary MATLAB code for sorting frequencies.

FIG. 34 may show exemplary MATLAB code for combining amplitude pairs.

FIG. 35 may show exemplary MATLAB code for reconstructing a time domain.

FIG. 36 may show exemplary time domain and frequency domain plots for arandom 25^(th) degree Taylor polynomial and its instantaneous spectrum.

FIG. 37 may show an exemplary comparison of ISA and FT spectral usage.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Aspects of the invention are disclosed in the following description andrelated drawings directed to specific embodiments of the invention.Alternate embodiments may be devised without departing from the spiritor the scope of the invention. Additionally, well-known elements ofexemplary embodiments of the invention will not be described in detailor will be omitted so as not to obscure the relevant details of theinvention. Further, to facilitate an understanding of the descriptiondiscussion of several terms used herein follows.

The word “exemplary” is used herein to mean “serving as an example,instance, or illustration.” Any embodiment described herein as“exemplary” is not necessarily to be construed as preferred oradvantageous over other embodiments. Likewise, the term “embodiments ofthe invention” does not require that all embodiments of the inventioninclude the discussed feature, advantage or mode of operation.

Further, many embodiments are described in terms of sequences of actionsto be performed by, for example, elements of a computing device. It willbe recognized that various actions described herein can be performed byspecific circuits (e.g., application specific integrated circuits(ASICs)), by program instructions being executed by one or moreprocessors, or by a combination of both. Additionally, these sequence ofactions described herein can be considered to be embodied entirelywithin any form of computer readable storage medium having storedtherein a corresponding set of computer instructions that upon executionwould cause an associated processor to perform the functionalitydescribed herein. Thus, the various aspects of the invention may beembodied in a number of different forms, all of which have beencontemplated to be within the scope of the claimed subject matter. Inaddition, for each of the embodiments described herein, thecorresponding form of any such embodiments may be described herein as,for example, “logic configured to” perform the described action.

According to an exemplary embodiment, a method for spiral polynomialdivision multiplexing (SPDM) may be described. In an exemplaryembodiment, a SPDM multiplexer may be based on amplitude modulation of aset of basis polynomials, or sub-channels, that cover all polynomials ofa given degree. According to an exemplary embodiment, SPDM sub-channelsmay not modulate sinusoidals. Instead, SPDM sub-channels may (amplitude)modulate component polynomials which are orthogonal to each other in thetime domain or in the polynomial coefficient space.

In an exemplary embodiment, the basis polynomials may be orthogonal inthe time domain; for example, in some embodiments, the basis set may bethe Chebyshev polynomials. In other embodiments, the basis polynomialsmay be orthogonal in the polynomial coefficient space; for example, insome exemplary embodiments, the basis set may be Cairns polynomials.

In an exemplary embodiment, the SPDM multiplexer (SPDM MUX or MUX) maycombine, or may be responsible for combining, independently-modulatedcomponent polynomials into a “message polynomial”, which may beconvolved with a “reference polynomial” to produce a “transmissionpolynomial”. The SPDM multiplexer may then generate a series ofamplitude values from the transmission polynomial, which may then betransmitted as a signal from a transmitter.

In some embodiments, bandlimiting of a transmission may be performed bymaking use of Instantaneous Spectral Analysis (ISA) to limit the rangeof frequencies used by expressing the signal in terms of sinusoidalswith continuously-varying amplitude. In an exemplary embodiment,transmissions may be limited to a range of frequencies equal to thereciprocal of the transmission duration.

According to an exemplary embodiment, upon receiving a transmissionpolynomial signal at a receiver, a SPDM demultiplexer (SPDM DMX or DMX)may then be configured to reverse the composition of the transmissionpolynomial in order to identify the sub-channel information.

In some exemplary embodiments, a SPDM DMX may use one of the followingapproaches. In a first approach, if the basis polynomials are Cairnspolynomials, a polynomial can be fit to the received amplitude values toreconstruct the transmission polynomial, then sub-channel amplitudes canbe determined by orthogonal projection in the polynomial coefficientspace. In a second approach, for the Cairns or other basis polynomialsets, a SPDM DMX may build templates for the amplitude sequencegenerated by all possible transmission polynomials, then apply astandard minimum distance metric to identify a received transmissionpolynomial, from which the sub-channel amplitudes can be determined bytable lookup. In other exemplary embodiments, another method of SPDMdemultiplexing may also be used, if desired.

Certain advantages may be understood for SPDM multiplexing as comparedto other multiplexing methods. For example, as compared to othermultiplexing methods in widespread use, SPDM may offer high spectralefficiency. The combination of SPDM with ISA may allow polynomials ofarbitrarily high degree to be transmitted with a very low bandwidthtransmission time or bandwidth delay product, such as a bandwidthtransmission time product (BT) equal to one. Since a polynomial ofdegree D is equivalent to D+1 independent amplitude values, this makesit possible to exceed the Nyquist rate (which arises from the samplingtheorem, proven under the assumption that sinusoidals have constantamplitude over an evaluation interval). Increasing the rate at whichindependent amplitude values can be transmitted using a fixed frequencyrange can be used in principle to push spectral efficiency tounprecedentedly high levels.

In some other exemplary embodiments, depending on configuration, otherpotential advantages of SPDM may include very precise and robustsynchronization and power normalization and a very large and flexiblewaveform design space that can be used to reject coherent interferenceor for other applications. In some exemplary embodiments, SPDM may notbe dependent on and may not require sinusoidal orthogonality.

For example, in an exemplary embodiment of a SPDM system, an advantageof the SPDM system may be that very precise time synchronization may bepossible with limited overhead in terms of bandwidth, power, and time bymaking use of the properties of polynomials. For example, in someexemplary embodiments, for example exemplary embodiments above certainnoise levels, less than 1% error may be achieved in a singleTransmission Time Interval (TTI). In some exemplary embodiments, forrelatively low-impairment channels, very precise time synchronizationmay be achieved without any reduction to traffic (user data throughput)or increase in bandwidth usage.

In another exemplary embodiment of a SPDM system, SPDM synchronizationmay be sufficiently robust that for one-to-one communication it opensthe possibility of deliberately desynchronizing the channel on a per-TTIbasis and using the overall channel timing as an additional modulationparameter, before the sub-channels are analyzed. This may furtherimprove throughput, if desired.

In another exemplary embodiment of a SPDM system, a SPDM system mayoffer precise and robust power normalization. An exemplary embodiment ofan SPDM system may allow for new techniques to be used to enforce PAPRlimits.

In another exemplary embodiment of a SPDM system, a SPDM system may havea huge and flexible waveform design space, since the basis polynomialscover the space of all polynomials of a given degree. In some exemplaryembodiments, waveform sets may be designed, based on this design space,specifically to reject coherent interference, or for other purposes, asmay be desired.

Lastly, in another exemplary embodiment of a SPDM system, an SPDM systemmay be implemented without the need for any frequency alignment orsinusoidal orthogonality requirements, which may be accomplished basedon the fact that SPDM is not based on sinusoidals.

However, some embodiments of SPDM may also be computationally intensive,in that they may benefit from sampling above the Nyquist rate andanalyzing the resulting data. In some exemplary embodiments, SPDMsystems may also be reliant on the accuracy of fitting a polynomial tonoisy amplitude values (for example, according to an exemplaryembodiment wherein the SPDM DMX uses a polynomial fit), and/or have highnumerical precision sensitivity. For example, in an exemplaryembodiment, SPDM may use Taylor polynomials, which have coefficientsproportional to the inverse factorial of the polynomial degree. Forhigh-degree polynomials, this may require accurately processing a verywide range of numerical magnitudes.

Mathematical Background

In the following section, a discussion of some of the mathematics thatmay serve as a basis for understanding a SPDM multiplexing anddemultiplexing system may be provided. Certain summaries and proofs ofthese mathematics may be available in other documents. See, for example,Jerrold Prothero, The Shannon law for non-periodic channels, AstrapiTechnical Report R-2012-1 (2012), available athttp://www.astrapi-corp.com/technology/white-papers/, and JerroldProthero, Euler's formula for fractional powers of I (2007), availableat http://www.astrapi-corp.com/technology/white-papers/, the contents ofwhich are incorporated by reference.

To summarize, SPDM may make use of polynomial coefficient projectiononto Cairns space as the basis for composing from and decomposing intosub-channels.

The familiar Euler's formula

e ^(it)=cos(t)+i·sin(t)  (1)

can be generalized by raising the imaginary constant i on the left sideto fractional powers. The new term

$\begin{matrix}e^{{ti}^{\; {(2^{2 - m})}}} & (2)\end{matrix}$

reduces to the standard Euler's term in the special case m=2. Table 1shows the generalized terms of Euler's formula for this and otherpositive integer values of m:

TABLE 1 Generalized Euler's Term as a Function of m m 0 1 2 3 4 5 . . .e^(ti^((2^(2 − m)))) e^(t) e^(−t) e^(it) e^(ti) ^((1/2)) e^(ti) ^((1/4))e^(ti) ^((1/8)) . . .

The standard Euler's formula, shown in equation (1), can be proved byexpanding e^(it) as a Taylor series and grouping real and imaginaryterms. The same procedure can also be used for the term in (2) to derivea generalization of Euler's formula for integer m≥0:

$\begin{matrix}{e^{{ti}^{\; {(2^{2 - m})}}} = {\sum\limits_{n = 0}^{{\lceil 2^{m - 1}\rceil} - 1}\; {i^{{n2}^{2 - m}}{\psi_{m,n}(t)}}}} & (3)\end{matrix}$

Where

$\begin{matrix}{{\psi_{m,n}(t)} = {\sum\limits_{q = 0}^{\infty}\; {( {- 1} )^{q \cdot {\lceil 2^{1 - m}\rceil}} \cdot \frac{t^{\; {{q \cdot {\lceil 2^{m - 1}\rceil}} + n}}}{( {{q \cdot \lceil 2^{m - 1} \rceil} + n} )!}}}} & (4)\end{matrix}$

The ψ_(m,n)(t) are called the “Cairns series functions”.

Notice that ψ_(2,0)(t) and ψ_(2,1)(t) give us the Taylor series for thestandard cosine and sine functions, respectively.

Each value of m produces a “level” of functions ψ_(m,n)(t). From thesummation limits in Equation 3, it can be seen that each level has atotal of [2^(m−1)] functions. Table 2 shows the progression of m versusthe number of functions at each level of m.

TABLE 2 Number of Functions at Each m-Level Level (m-value) 0 1 2 3 4 5. . . Number of functions 1 1 2 4 8 16 . . .

An important property of the ψ_(m,n)(t) is the regular pattern of theircoefficients. This regular pattern is shown in Table 3.

TABLE 3 The Cairns Series Coefficients 1 t $\frac{t^{2}}{2!}$$\frac{t^{3}}{3!}$ $\frac{t^{4}}{4!}$ $\frac{t^{5}}{5!}$$\frac{t^{6}}{6!}$ $\frac{t^{7}}{7!}$ . . . ψ_(0,0)(t) = e^(t) 1 1 1 1 11 1 1 . . . ψ_(1,0)(t) = e^(−t) 1 -1 1 -1 1 -1 1 -1 . . . ψ_(2,0)(t) =cos(t) 1 0 -1 0 1 0 -1 0 . . . ψ_(2,1)(t) = sin(t) 0 1 0 -1 0 1 0 -1 . .. ψ_(3,0)(t) 1 0 0 0 -1 0 0 0 . . . ψ_(3,1)(t) 0 1 0 0 0 -1 0 0 . . .ψ_(3,2)(t) 0 0 1 0 0 0 -1 0 . . . ψ_(3,3)(t) 0 0 0 1 0 0 0 -1 . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Here, the rows indicate the Taylor series coefficients for eachψ_(m,n)(t). For instance, the row for ψ_(2,0)(t)=cos(t) indicates that

$\begin{matrix}{{\cos (t)} = {1 - \frac{t^{\; 2}}{2!} + \frac{t^{\; 4}}{4!} - \frac{t^{\; 6}}{6!} + \ldots}} & (5)\end{matrix}$

Table 3 shows that the ψ_(m,n)(t) coefficients may define a set oforthogonal vectors. More precisely, if M is a positive integer, then thevectors formed from the first 2^(M) coefficients of the functionsψ_(m,n)(t) for 0≤m≤M may constitute a set of orthogonal basis vectorsfor a 2^(M)-dimensional space. These can be normalized to produce theorthonormal “2^(M) Cairns basis vectors”.

The existence of the 2^(M) Cairns basis vectors implies that any Taylorpolynomial P of degree k<2^(M) can be orthogonally projected ontopolynomials formed from the first 2^(M) terms of the Cairns seriesfunctions simply by taking the inner product of P's coefficients withthe 2^(M) Cairns basis vectors. The resulting coefficients for eachCairns basis function are referred to as the “projection coefficients”.

The first 2^(M) terms of the Cairns series functions ψ_(m,n)(t) are onlyan approximation to the full infinite series expansion of theψ_(m,n)(t). However, the error in the approximation is O(t⁽² ^(M) ⁾),with a reciprocal factorial coefficient, and therefore falls off veryrapidly as M increases. For high-degree polynomials, therefore, it isreasonable to speak of projecting onto the ψ_(m,n)(t) by this procedure.

It is well-known that the cosine and sine functions of Euler's formulacan be represented not only by Taylor series but also by sums of complexexponentials.

Explicitly:

$\begin{matrix}{{{{\cos (t)} = {{1 - \frac{t^{\; 2}}{2!} + \frac{t^{\; 4}}{4!} - \frac{t^{\; 6}}{6!} + \ldots} = {\frac{1}{2}( {e^{it} + e^{- {it}}} )}}}{and}}\mspace{461mu}} & (6) \\{{\sin (t)} = {{t - \frac{t^{\; 3}}{3!} + \frac{t^{\; 5}}{5!} - \frac{t^{\; 7}}{7!} + \ldots} = {\frac{1}{2i}( {e^{it} - e^{- {it}}} )}}} & (7)\end{matrix}$

This characteristic also holds for the generalized Euler's formula. Forexample, Equation (8) can be defined as the following:

$\begin{matrix}{{E_{m,n}(t)} = {\frac{1}{\lceil 2^{m - 1} \rceil}{\sum\limits_{p = 0}^{{\lceil 2^{m - 1}\rceil} - 1}{i^{{- {n{({{2p} + 1})}}}2^{2 - m}}e^{{ti}^{{({{2p} + 1})}2^{2 - m}}}}}}} & (8)\end{matrix}$

Where the E_(m,n)(t) are called the “Cairns exponential functions.”

By expanding the right side of Equation 8 as a sum of Taylor polynomialsand recursively cancelling terms, it can be shown that for all m and n

E _(m,n)(t)=ψ_(m,n)(t)  (9)

Equation 9 tells us that once a polynomial has been projected onto theCairns series functions, it can be immediately converted into a sum ofcomplex exponentials. In some exemplary embodiments, this may be used toconvert a polynomial into sinusoidals with continuously-varyingamplitude by an SPDM MUX.

As mentioned above, the Cairns basis functions allow any Taylorpolynomial (which includes any polynomial with positive integercoefficients) to be projected onto the Cairns series functions.Conversely, any weighted sum of Cairns series functions correspondsuniquely to a particular Taylor polynomial. In some exemplaryembodiments, this principle may be used by an SPDM MUX and DMX tocompose and decompose sub-channels. According to one exemplaryembodiment of SPDM, each sub-channel may correspond to oneamplitude-modulated Cairns function ψ_(m,n)(t). Other exemplaryembodiments of SPDM using different principles may also be understoodand may be used as desired.

According to an exemplary embodiment, SPDM may be based on the Cairnsfunctions; it may thus be worthwhile to examine the properties of Cairnsfunctions in further detail. As the Cairns functions at level m aregenerated from the generalized Euler's term

e^(ti^( (2^(2 − m)))),

this generalized term may be examined specifically in more detail.

By using the identity

e ^(iπ/2) =i  (10)

which is a special case of Euler's formula, it follows that

$\begin{matrix}{e^{{ti}^{\; {(2^{2 - m})}}} = {e^{t \cdot {\cos {({\pi 2}^{1 - m})}}}e^{i \cdot t \cdot {\sin {({\pi 2}^{1 - m})}}}}} & (11)\end{matrix}$

On the right side of Equation (11), the first factor, e^(t·cos(π2)^(1−m) ⁾, is a real-valued exponential; the second factor,e^(i·t·sin(π2) ^(1−m) ⁾, describes a circle in the complex plane. Theproduct of the two factors describes a spiral in the complex plane.

In Equation (11) above, m=0 generates the standard rising exponential;m=1 generates the standard falling exponential; and m=2 generates thestandard sine and cosine functions.

From Equation (11), we can see that as m increases above m=2, the rateof growth increases and the rate of rotation decreases. In the limit as

m → ∞, e^(ti^( (2^(2 − m))))

converges back to e^(t).

At every level of m, the ψ_(m,n)(t) share the growth and frequencyproperties of their generating term

e^(ti^( (2^(2 − m)))).

For instance, the Cairns functions with the fastest rotation (andslowest growth) occur at m=2.

Another important observation is that for m≥2 Cairns functions with evenn are always symmetric around t=0, and Cairns functions with odd n arealways anti-symmetric around t=0.

The Cairns series polynomials through m=3 are plotted in FIGS. 1-8 forthe interval −3.1≤t≤3.1. For example, FIG. 1 shows an exemplary plot ofa Cairns function for m=0 and n=0. FIGS. 3-8 demonstrate the observationdiscussed above; for example, FIG. 3, showing the exemplary plot of aCairns function for m=2 and n=0, is symmetric about t=0, while FIG. 4,showing the exemplary plot of a Cairns function for m=2 and n=1, isanti-symmetric about t=0.

In some exemplary embodiments, the polynomials may be truncated at asufficiently small value so as not to adversely affect accuracysignificantly. For example, according to the exemplary embodiments shownin FIGS. 1-8, the polynomials may be truncated to 7^(th) degreepolynomials, for which they provide a spanning set, consistent with theabove discussion. In some exemplary embodiments, the polynomials may beconstructed and plotted with eight points, uninterpolated, forconsistency with SPDM techniques. According to an exemplary embodiment,SPDM may not smooth through interpolation until the transmissionpolynomial is formed by convolving a sum of Cairns functions with areference polynomial.

The SPDM Instantiations

According to some exemplary embodiments, SPDM may be built on the ideaof amplitude modulation of basis polynomials, which may then betransmitted using ISA to limit bandwidth usage.

According to an exemplary embodiment, SPDM may be implemented byamplitude modulating a set of basis polynomials which are orthogonal inthe time domain, such as the Chebyshev polynomials of the first kind, toproduce a transmission polynomial. In an embodiment, a sequence ofamplitude values generated by the transmission polynomial may betransmitted using ISA. The receiver may then be configured to detect thetransmission polynomial using, for example, a standard minimum Euclideandistance technique. In an embodiment, the basis polynomial amplitudesmay then be determined by table lookup.

In another exemplary embodiment, SPDM may be implemented such that itmakes use of another method of composing and decomposing the basispolynomials, for example orthogonality in the polynomial coefficientspace, using Cairns polynomials. This and other embodiments may bedescribed further on.

According to an exemplary embodiment, the SPDM algorithm may function asfollows. In an embodiment, each sub-channel may correspond to a Cairnsfunction selected from 0≤m≤M, where 2^(M) is the number of sub-channels.All sub-channels may be parameterized for the same “evaluation interval”symmetric around t=0. All sub-channels may be normalized to have thesame RMS power budget over the evaluation interval. Synchronization andpower normalization may be achieved between the transmitter andreceiver, using techniques described in a later section. Information maybe loaded onto each sub-channel through amplitude modulation of theCairns functions based on input bit sequences. For example, according toone exemplary embodiment, if a given sub-channel conveys two bits pertransmission, it may have four possible amplitude values, correspondingto the bit sequences 00, 01, 10, and 11 (with appropriate Gray coding).

According to an exemplary embodiment, since amplitude modulation of theCairns functions provides a spanning set for all polynomials of a givendegree, phase may not be considered, as no loss of generality may occurfrom not considering other possible modulations such as phase.

In an exemplary embodiment, the sub-channels may be summed according totheir amplitude modulation to produce a composite “message polynomial”.Optionally, according to some exemplary embodiments, the messagepolynomial may be multiplied (convolved) with a “reference polynomial”.The reference polynomial may be introduced in order to, for example,provide a known pattern to look for, or in order to better characterizea channel. In another embodiment, the reference polynomial may beintroduced in order to, for example, provide a smooth join betweenTTI's; alternatively, or additionally, this may be accomplished throughISA interpolation. In some embodiments, a reference polynomial designedto provide smoothing and to reduce the amplitude to zero at the TTIboundaries may be any Cairns function with m≥2, n=0 in the region aroundt=0, if scaled and translated to have amplitude values in the range zeroto one. According to some embodiments, the evaluation region may thus bepicked to be symmetric around t=0.

According to an exemplary embodiment, the simplest reference polynomialwith reasonable properties may be ψ_(3,0)(t), truncated to fourthdegree: 1−t⁴/4! In some embodiments, this may thus be used as thedefault reference polynomial for SPDM. In other embodiments, largervalues of m in ψ_(m,0)(t) may be used, which may provide better roll-offproperties, but require higher-degree polynomials. In an exemplaryembodiment, higher degree reference polynomials may effectively reducethe degrees of freedom available to the message polynomial, andtherefore reduce traffic (user data throughput). In some exemplaryembodiments, the reference polynomial may be known ahead of time to boththe transmitter and receiver.

The product of the message polynomial with the reference polynomial istermed the “transmission polynomial”. (If there is no referencepolynomial, the message polynomial is also termed the transmissionpolynomial.) In an embodiment, the degree of the transmission polynomialis the sum of the degrees of the message and reference polynomials. Thetransmission polynomial may then be converted into a sum of sinusoidalswith continuously-varying amplitude using ISA.

In an embodiment, a sequence of amplitude values over the evaluationinterval may be generated from the ISA representation of thetransmission polynomial and may be transmitted, which may in anexemplary embodiment constitute the data for one TTI. In someembodiments, the number of points generated may be at least one greaterthan the degree of the transmission polynomial. However, values inexcess of this may also be used; the amplitude values may be upsampledbeyond this point, for example in order to ensure smooth interpolation,or so that the DMX has more points to work with as explained below.According to an exemplary embodiment, which may for ease of reference beused in subsequent examples, the transmission polynomial may beupsampled by a factor of eight.

According to an exemplary embodiment, in the receiver, the DMX may fit apolynomial to the received data, regenerating the transmissionpolynomial. Since the default reference polynomial pushes amplitudevalues at the TTI boundary to zero (which are therefore noise-prone),according to an exemplary embodiment, the receiver may be aided byupsampling in the transmitter, which may facilitate regeneration of thetransmission polynomial from high-amplitude points in the middle of thetransmission.

According to an exemplary embodiment, the DMX may deconvolve thereference polynomial from the transmission polynomial to recover thetransmitted message polynomial.

According to an exemplary embodiment, the sub-channel amplitudes maythen be re-constructed by projecting the message polynomial onto Cairnsspace.

According to an exemplary embodiment, the sub-channels may then beconverted back into bits by mapping the amplitude values for eachsub-channel into bit sequences.

FIGS. 9 and 10 may show two sample transmissions. In FIGS. 9 and 10,amplitudes generated by the transmission polynomial are shown as adotted (rather than dashed or solid) line; this may correspond to whatmay be sent over the channel. According to the embodiments shown inFIGS. 9 and 10, every TTI may start and end at amplitude zero due to theeffect of the reference polynomial.

The message polynomial (which may be obtained in the DMX by deconvolvingthe reference polynomial from the transmission polynomial) is shown inblack. The dashed black lines show the eight Cairns functions (selectedfrom m≤3) which may, when summed, produce the message polynomial. Theymay be obtained from the message polynomial by projection onto Cairnsspace, and their respective amplitudes may determine the sub-channelmessages.

According to the embodiments depicted in FIGS. 9 and 10, thetransmission and message polynomials may essentially overlap in thecenter of the TTI, where the reference polynomial has high amplitude. Atthe edges of the TTI, in an exemplary embodiment, the referencepolynomial may force the transmission polynomial to zero amplitude,which may cause it to diverge from the message polynomial.

Reference Polynomials

According to an exemplary embodiment, the purpose of the referencepolynomial may be as described above.

A feature of reference polynomials may be that deconvolution of thereference polynomial from the correct transmission polynomial in thereceiver allows the original message polynomial to be reconstructedunambiguously.

As indicated above, a low-degree reference polynomial with usefulproperties may be ψ_(3,0)(t) truncated to fourth degree. This referencepolynomial is scaled and vertically translated so that over theevaluation interval it has amplitudes in the range between 0 and 1. Sucha polynomial is depicted in FIG. 11.

According to an exemplary embodiment, the degree of the referencepolynomial may be kept as low as possible. This may be, for example,because the number of independent points a TTI must be capable ofrepresenting is equal to the degree of the transmission polynomial plusone, which is equal to the sum of the degrees of the message polynomialand the reference polynomial plus one. For example, in an exemplaryembodiment where the number of independent points the channel is capableof transmitting in a TTI is fixed, every degree in the referencepolynomial may effectively “steal” a degree from the message polynomial,and therefore may reduce traffic. Essentially, reference polynomialdegree in SPDM may impose a cost similar to guard bands in traditionalsignal modulation techniques.

Since the reference polynomial imposes a cost associated with itsdegree, in some exemplary embodiments, the degree of the messagepolynomial may be increased or maximized in order to reduce the relativeburden of the reference polynomial, and therefore increase traffic. Forinstance, according to some exemplary embodiments, the M=3 examples usedin this document may be moved to M=4 or higher.

However, in other exemplary embodiments, the degree of the messagepolynomial may not be increased or maximized, for one of severalreasons. For example, in an exemplary embodiment, increasing the valueof M may increase the computational burden on the system, as highervalues of M may require the SPDM MUX and DMX to generate and analyzehigher-degree polynomials. Increasing the value of M may also requirethat the complexity of the system be increased. Since SPDM may useTaylor polynomials, terms may have coefficients inversely proportionalto the factorial of each term's degree. High-degree polynomialstherefore may force the SPDM implementation to handle a very wide rangeof numbers with precision, increasing complexity.

Synchronization and Power Control

According to an exemplary embodiment, for the receiver to correctlyinterpret the information the transmitter provides, the two ends of thecommunication channel may have to agree on when a message starts. Theprocess for ensuring that this occurs is call “synchronization”.

Further, according to such an exemplary embodiment, the two ends of thecommunication channel may have to agree on the power of the transmittedmessage so that amplitude information is interpreted correctly. This maybe referred to as “power control”.

This section describes how synchronization and power control may bejointly performed by SPDM.

SPDM may support at least three methods for synchronization, the choiceof which may depend on how much of the channel power budget is divertedfrom traffic to synchronization in a particular TTI. According to anexemplary embodiment, the amount of power diverted from traffic will beminimized, since the purpose of the communication channel is to conveyuser information and since diverting power from traffic conflicts withthat goal.

According to an exemplary embodiment, very accurate synchronization maybe possible with no reduction in traffic for channels with relativelylow impairment.

In some exemplary embodiments, SPDM synchronization may be implementedsubject to one or more conditions. The below subsections describe SPDMsynchronization assuming that one of the following three conditionsholds:

Full Power Synchronization. The full channel power is made available forsynchronization in a particular TTI.

Limited Power Synchronization. Only the channel power associated withthe rising and falling exponential sub-channels is made available forsynchronization in a particular TTI.

No Power Synchronization. No channel power is made available forsynchronization.

Full Power Synchronization

According to an exemplary embodiment, when full power synchronization isused, SPDM may transmit for one TTI a message polynomial whosesub-channels include a rising and falling exponential with equal peakamplitude together absorbing all available channel power, with all othersub-channels set to amplitude zero. (According to an exemplaryembodiment, the transmission interval may be parameterized to besymmetric around t=0, which may result in the peak amplitude for therising and falling exponentials being equal if the two exponentials havethe same projection coefficient.) As shown below, this may provide thereceiver with a very strong signature for correct synchronization. (Theexponentials may, however, be bandlimited by ISA prior to transmission.)

According to an exemplary embodiment, the SPDM can use a brute-forcesynchronization strategy in which it tries all possible synchronizationphases (time offsets) until it identifies the correct synchronizationsignature.

On a channel with no impairments, FIG. 12 shows an exemplary embodimentof what the receiver may observe when it has correct synchronization.Also shown, in FIGS. 13 and 14, are exemplary embodiments of what thereceiver may see if it is off on the phase by only 1 part in 89;specifically, FIG. 13 depicts a phase mismatch of −1/89, while FIG. 14depicts a phase mismatch of +1/89. Put differently, the “mismatch”figures may show what the receiver would have to mistake for the correctphase in order to make a phase error of 1 part in 89. (In the exemplaryembodiments depicted in FIGS. 12 through 14, the “1 part in 89” phaseerror may be a reflection of the number of interpolation points chosenfor the transmission polynomial, and hence the number of amplitudevalues in the TTI. In some cases, synchronization could be achieved moreaccurately than this, depending on channel conditions. In some exemplaryembodiments, a different number of interpolation points may be chosen,if desired, which may result in differences in synchronizationaccuracy.)

Notably, the exemplary embodiments of “mismatch” message polynomialsdepicted in FIGS. 13 and 14 are quite different in shape from thecorrectly synchronized polynomial, despite differing in phase by only 1part in 89. Notice also that the vertical axes differ by more than afactor of ten.

The synchronization signature is so strong because a small displacementin phase may cause either the rising or falling exponent (the twocomponents in the synchronization message polynomial) to dominate overthe other, breaking the balance between them that is the signature ofcorrect synchronization. In physics terms, this balance between therising and falling exponentials may be understood as a deliberatelycreated unstable equilibrium, similar to balancing a pencil on its tip,which may be destroyed by even very small phase errors.

At the synchronization phase, the common received amplitude of therising and falling exponentials can be used for power normalization, bycomparing their observed amplitudes to a reference value.

According to an exemplary embodiment, the SPDM receiver may beconfigured to identify the correct synchronization phase by examiningpossible phase shifts of the received amplitude data, and finding themessage polynomial and message polynomial sub-channel coefficientsimplied by that phase. In an embodiment, the SPDM receiver may selectthe phase that produces the smallest Euclidean distance between thecorrect sub-channel coefficients and the observed sub-channelcoefficients as the correct phase for synchronization. For the aboveexample, the Euclidean distances of all possible synchronization phasesare plotted in FIG. 15.

According to the embodiment depicted in FIG. 15, the “correct”synchronization phase of 32/89 in this case has a significantly lowerEuclidean distance from the correct match than any other potentialsynchronization phases. Furthermore, the Euclidean distance errormeasures for adjacent phases have combined slopes on both sides thatpoint to the correct synchronization phase.

Limited Power Synchronization

The “full power” synchronization method depicted above may produce avery strong signature for synchronization in one TTI. However, it mayalso require using all communication power in that TTI forsynchronization, which may cut traffic to zero. According to anexemplary embodiment, however, a device may be synchronized accuratelywithout shutting down user communication for one TTI through the use oflimited power synchronization.

According to an exemplary embodiment, limited power synchronization maybe achieved by taking advantage of the fact that the rising and fallingexponents are orthogonal to all other Cairns functions in the messagepolynomial coefficient space.

According to such an embodiment, the limited power synchronizationmethod may operate as follows. First, a power budget may be assigned tothe rising and falling exponentials; for instance, in one exemplaryembodiment, each of the rising and the falling exponentials could begiven the same RMS power budget as all other Cairns function messagepolynomial sub-channels. Second, on a synchronization TTI, the risingand falling exponentials may both be transmitted with their maximumallowable power. In an exemplary embodiment, all other sub-channels canbe independently amplitude modulated as usual to carry traffic. Third,in the receiver, synchronization may proceed as with the “full power”method described above, with the distinction that only the rising andfalling exponent coefficients may be matched. In an embodiment, all ofthe other sub-channels may be handled using the “no power” approachdescribed below. In an exemplary embodiment, synchronization may use a“full power” approach using the rising and falling exponentcoefficients, a “no power” approach using other sub-channels, or acombination of the two; for example, synchronization may determine thebest combination of the “full power” approaches performed on somesub-channels and the “no power” approaches performed on othersub-channels, such as a transmission polynomial that produces the lowesttotal deviation from both sets of sub-channels.

For example, according to an exemplary embodiment, the rising andfalling exponential coefficients may be assumed to be set to a specificfraction of the available signal power, such as 1/8 of the signal powereach if there are 8 basis polynomials or equivalently 8 sub-channels.The remaining sub-channels may be assumed to have coefficients of zero.If there are 8 sub-channels, this means that user data transmission maybe reduced by 2/8=1/4 by synchronization.

According to an exemplary embodiment, “limited power” synchronizationmay be less robust than “full power” synchronization. This may bebecause the power given to the rising and falling exponentials issmaller, and because it is not possible to assume that non-exponentialamplitudes should be zero as part of the match signature. However,“limited power” synchronization may still be adequately robust for manyimplementations.

FIGS. 16-19 may mirror FIGS. 12-15, discussed above; however, FIGS.16-19 may show the altered performance of a system that is using“limited power” synchronization rather than “full power”synchronization. On a channel with no impairments, FIG. 16, like FIG.12, shows what the receiver may observe when it has correctsynchronization. Also shown, in FIGS. 17 and 18, are exemplaryembodiments of what the receiver may see if it is off on the phase byonly 1 part in 89; specifically, FIG. 17 depicts a phase mismatch of−1/89, while FIG. 18 depicts a phase mismatch of +1/89. Put differently,the “mismatch” figures may show what the receiver would have to mistakefor the correct phase in order to make a phase error of 1 part in 89.FIG. 19 may, like FIG. 15, show the Euclidean distances of all possiblesynchronization phases.

No Power Synchronization

According to another exemplary embodiment, it may be possible under somecircumstances to synchronize SPDM without dedicating any power tosynchronization, thus allowing full traffic to take place duringsynchronization. According to such an embodiment, this may be done bytaking advantage of the fact that the SPDM message polynomial space ishuge (it spans all polynomials up to degree 2^(M)−1), but that most ofthe polynomials in the SPDM message polynomial space would not be ableto have been generated by the SPDM transmitter, for example due to poweror PAPR limitations. If the synchronization phase is incorrect, thereceived message polynomial is highly likely to contain componentamplitude values that the transmitter could not have generated.

As such, according to such an embodiment, the receiver can takeadvantage of this fact to synchronize on the phase where the messagepolynomial component amplitudes are closest to all being zero (theaverage value of component amplitudes for allowable random messagepolynomials).

However, two potential concerns exist regarding “no power”synchronization. First, according to some exemplary embodiments, “nopower” synchronization may fail to consistently transmit data even withno channel impairments, because a compliant message polynomial could begenerated essentially by chance at an incorrect synchronization phase.In some embodiments, the receiver may take action to mitigate this; forexample, in an exemplary embodiment, the receiver may choose or may beable to choose from several potential phases based on those phaseshaving message polynomial component amplitudes that are all closest tozero. In an exemplary embodiment, the receiver may then parse thecontent of each of these potential messages and select the one that ismost likely to have been sent; for example, if one such potentialmessage contains a working image file or a number of words recognized inthe receiver's dictionary, and all of the other potential messages wouldbe meaningless noise, the receiver could identify the message containingthe working file or the recognized words as being the most likely one tohave been sent to the receiver and thus the proper message to receive.Alternatively, each of the potentially valid messages may be presented,if desired.

According to some embodiments, “no power” synchronization also may notbe used to normalize power by the method used in “full power” and“limited power” synchronization, because “no power” synchronization doesnot provide a known reference amplitude value. However, in an exemplaryembodiment, power normalization may still be performed or may beperformable; for example, in an exemplary embodiment, powernormalization may be performed by examining the maximum amplitudesobserved in the sub-channels, and comparing them to the referencemaximum amplitude values for the sub-channels.

However, according to some embodiments, “no power” synchronization maystill be used effectively, thus allowing full traffic duringsynchronization. FIG. 23 shows a plot of No Power SynchronizationDistance by Phase, in which the correct phase is 43/89; this may becompared to a similar plot for “full power” synchronization, shown inFIG. 15 (correct phase 32/89). FIG. 23 indicates that thesynchronization signature for “no power” synchronization may, in someembodiments, be almost as clear as for the “full power” synchronizationmethod discussed earlier, which may make “no power” synchronizationpractical for use.

According to some embodiments, a system making use of “no power”synchronization may also be able to take advantage of the fact that,when the synchronization phase is wrong, the message polynomial willgenerally not be close to the transmission polynomial in the center ofthe TTI if the default reference polynomial is used, as may be the casefor correct synchronization phase alignment. According to thoseembodiments, the message polynomial will generally be close to thetransmission polynomial in the center of the TTI in the event that thecorrect synchronization phase alignment is identified.

Turning now to exemplary FIGS. 20-23, FIGS. 20-23 may mirror FIGS.12-15, discussed above; however, FIGS. 20-23 may show the alteredperformance of an exemplary embodiment of a system that is using “nopower” synchronization rather than “full power” synchronization. On achannel with no impairments, FIG. 20, like FIG. 12, shows an exemplaryembodiment of what the receiver may observe when it has correctsynchronization. Also shown, in FIGS. 21 and 22, are exemplaryembodiments of what the receiver may see if it is off on the phase byonly 1 part in 89. Specifically, FIG. 21 depicts an exemplary embodimentof a received signal having a phase mismatch of −1/89, while FIG. 22depicts an exemplary embodiment of a received signal having a phasemismatch of +1/89. Put differently, the “mismatch” figures may show whatthe receiver would have to mistake for the correct phase in order tomake a phase error of 1 part in 89. FIG. 23 may, like FIG. 15, show theEuclidean distances of all possible synchronization phases.

Projection Coefficient Rotation and Dimension Reduction

When using the Cairns polynomials as the basis polynomials, the set ofprojection coefficients can be thought of as forming a multi-dimensionalvector space.

For instance, according to an exemplary embodiment wherein Cairnspolynomials are used as the basis polynomials in an SPDM system, a SPDMmay make use of the first 8 Cairns functions available through the m=3level: ψ_(0,0)(t), ψ_(1,0)(t), ψ_(2,0)(t), ψ_(2,1)(t), ψ_(3,0)(t),ψ_(3,1)(t), ψ_(3,2)(t), and ψ_(3,3)(t). Any polynomial of degree 7 orlower may correspond uniquely to some set of 8 real-valued coefficientsfor these functions, which may be obtainable by projecting thepolynomial onto Cairns space. The values of the 8 projectioncoefficients can be thought of as forming a vector in an 8-dimensionalspace. The 2^(M=3) Cairns basis vectors provide a set of orthonormalaxes spanning this space. Such a configuration of projectioncoefficients may allow mathematical tools developed for dealing withvector spaces to be applied to the Cairns basis vectors.

For example, the idea of axial rotation may be applied to the vectorspace. A set of axes may be called a “reference frame”. A referenceframe can be rotated to make certain problems easier to analyze, forinstance to align a symmetry in the data with the axes. It is alsopossible to reduce the dimensionality of a vector space by removing oneor more axes, either in the original or in the rotated reference frame.This prevents vectors from being formed that are wholly or partiallyaligned with the removed axis or axes. In some exemplary embodiments,this technique may be applied to rejecting coherent interference and toreducing PAPR.

Coherent Interference Rejection

As with message polynomials, according to an exemplary embodiment,coherent interference may be projected onto Cairns space for analysis.In some embodiments, to the extent that a clear pattern for the channelimpairment appears, an SPDM system may be configured to adjust itself toimprove performance.

To illustrate how this may be implemented, according to an exemplaryembodiment, for an interfering signal, an SPDM system may fitpolynomials to the interfering signal over each of a series of TTI's,and project each of the resulting series of polynomials onto Cairnsspace for m≤3. In an example, the interfering signal over the series ofprojections may be supposed to be largely characterized by a1-dimensional pattern of variation in the projection coefficient space,though in some exemplary embodiments higher-dimensional cases can alsobe handled, if desired.

In this case, the interference can be viewed as occurring along a vectorin the projection coefficient space. In an embodiment, the projectioncoefficient reference space may be rotated, as described in the previoussection, so that the interference vector falls along one axis.

In an embodiment, a SPDM system can then remove the interference vectorby dropping the dimensionality of the projection coefficient space by 1,using only the remaining dimensions to generate message polynomials.Since the remaining dimensions are orthogonal to the interferencevector, according to an exemplary embodiment, the influence of thecoherent interference may be minimized.

In an exemplary embodiment, because this technique may reduce the numberof sub-channels, the power budget per channel may be proportionatelyincreased in order to improve the information capacity per sub-channel.

PAPR Control

According to an exemplary embodiment, it may be desirable to have a SPDMsystem have a low average PAPR (peak-to-average power ratio) or lowworst-case PAPR. In some embodiments, this may be difficult to achieve;since the 2^(M) Cairns basis functions on which SPDM is based span allpolynomials of degree 2^(M)−1 or less, summing the independent amplitudemodulation of Cairns functions can sometimes produce polynomials withvery poor PAPR performance.

According to some embodiments, PAPR may be adjusted downward byapplication of the default reference polynomial. However, according toan exemplary implementation of SPDM, PAPR may only be adjusted downwardif the message polynomial peak appears towards the ends of the TTI,where the reference polynomial pulls down amplitudes. This may beparticularly useful for taming the rising and falling exponentials.However, in the center of the TTI, the reference polynomial may bydesign have as little effect as possible, so that the maximum power isavailable for traffic. As such, alternative solutions for reducing PAPR,instead of or in addition to making use of the reference polynomial, mayalso be considered, as desired.

A fundamental problem for SPDM PAPR control is that the messagepolynomial space is very rich, and includes polynomials that, in someexemplary embodiments, may be undesirable to generate. In an exemplaryembodiment, this may be solved by restricting the message polynomialspace in some way. According to some exemplary embodiments of SPDM PAPRcontrol, three methods to do so may be envisioned: paired Cairnsfunction modulation; message re-mapping; and projection coefficientrotation. These techniques may be combined, or other techniques may alsobe used, as desired.

According to a first exemplary embodiment, paired Cairns functionmodulation may be used. According to one exemplary embodiment, thedomain of allowable projection coefficients for the Cairns functions maybe limited by transitioning from amplitude modulation of individualCairns functions, as described above, to amplitude modulation of pairsof Cairns functions with approximately opposing growth patterns. Forinstance, in an exemplary embodiment of a SPDM system, instead ofmodulating ψ_(3,0)(t), ψ_(3,1)(t), ψ_(3,2)(t), and ψ_(3,3)(t) asindependent sub-channels, one can modulate the sumsψ_(3,0)(t)+ψ_(3,2)(t) and ψ_(3,1)(t)−ψ_(3,3)(t) (see generally FIGS.1-8). While this approach may reduce modulation degrees of freedom, itmay not reduce traffic as improved PAPR performance may increase usableRMS power, allowing for more amplitude levels in the remainingsub-channels. According to some exemplary embodiments of SPDM systems,since SPDM has a huge polynomial design space available, a SPDM systemmay be configured to trade polynomial range for improved signalproperties.

According to another exemplary embodiment, message re-mapping may beused. In an embodiment, for a given configuration of evaluationinterval, available sub-channels, and available amplitude levels persub-channel, it may also be possible to pre-generate all possiblemessage polynomials and review their PAPR characteristics. Messagepolynomials with unacceptable PAPR performance could then be identifiedin advance, and limited or eliminated as necessary. A number ofpotential options for limiting or eliminating polynomials with bad PAPRexist.

For example, according to one exemplary embodiment, bad PAPR messagepolynomials may be prevented from being specified at run-time. However,this may impose a cross sub-channel constraint and therefore preventindependent transmission on the sub-channels, which may be undesirablein some implementations. It may also make SPDM unusable for many-to-onecommunication.

However, according to a second embodiment applicable to one-to-onecommunication channels, a transmitter may allow the sub-channels tocollectively specify an unacceptable message polynomial, but then map itto an acceptable message polynomial which the transmitter and receiverhave agreed in advance will be treated as equivalent. This may support avirtual sub-channel space in which each sub-channel can be modulatedcompletely independently of the rest, combined with a physical channelthat is guaranteed to maintain certain PAPR performance requirements. Inan embodiment, a certain amount of the sub-channel space with good PAPRperformance may have to be reserved to support the re-mapping. However,again, an SPDM system may be configured to trade polynomial design spacefor improved signal properties.

According to another exemplary embodiment, projection coefficientrotation may be used. Such a method may involve using a rotationaltechnique, such as the rotational technique discussed in a previoussection, to reduce PAPR by removing high-peak Taylor terms from themessage polynomial generation space.

For a given joint specification of M (which, aside from determining thenumber of Cairns functions, also determines the highest degree Taylorterm in use), the evaluation interval, and the reference polynomial,there will be some particular positive integer n_(p) for which theTaylor term t^(n) ^(p) /n_(p)! produces a higher peak value than anyother t^(n)/n! over the evaluation interval. This term t^(n) ^(p) /n_(p)! is the biggest PAPR offender; according to an exemplary embodiment, aSPDM system may thus reduce the impact of this term or may prevent thisterm from being generated.

If n_(p) is known, according to an exemplary embodiment, t^(n) ^(p)/n_(p) may be treated as a single-term message polynomial, and may beprojected onto Cairns space. This provides the projection coefficientsfor the Cairns functions that collectively generate t^(n) ^(p) /n_(p).In an embodiment, these coefficients may then be interpreted as a vectorand a technique such as the technique described in a previous sectionmay then be used in order to ensure that t^(n) ^(p) /n_(p) will not begenerated.

Depending on the PAPR requirements, in some exemplary embodiments,removing only t^(n) ^(p) /n_(p) may not be sufficient. In some exemplaryembodiments, it may be necessary or desirable to remove one or moreadditional terms as well. This can be accomplished by repeating therotation and dimension reduction procedure.

In some exemplary embodiments, it may not be desirable to reduce theprojection coefficient space dimensionality below some point; forexample, in an exemplary embodiment, a certain number of independentsub-channels may be necessary or desirable to support how thecommunication channel is used. According to said exemplary embodiments,a single composite vector which is a weighted sum of vectorscorresponding to Taylor terms that it may be desirable to remove may beformed. This composite vector may then be removed from the Cairnsprojection space. In an embodiment, removing the composite vector fromthe Cairns projection space may reduce the maximum value producible by amessage polynomial (and therefore the PAPR), although not assignificantly as if the vectors summed into the composite vector wereall removed individually.

In one exemplary embodiment, the term n_(p) corresponding to the termt^(n)/n! with the maximum peak value over the evaluation interval may befound by individually numerically evaluating all terms t^(n)/n!, for0≤n<2^(M), over the evaluation interval, after convolving eachindividually with the reference polynomial. In some exemplaryembodiments, this may not be particularly computationally expensive; insome exemplary embodiments, this may even be done during system design,rather than at run-time, if desired. In other exemplary embodiments,another technique may be used.

A discussion of which Taylor terms may, in some exemplary embodiments,be problematic may be provided. The nature of Taylor terms is that theyare “weighed down” by their reciprocal factorial coefficients, so thatin the interval close to zero the terms with lower degree will dominatehigher-degree terms. However, further from zero the more rapid growth ofthe higher-degree terms dominates. A transition point between thesegrowth terms may be identified.

To formalize this question, one can ask for the value of t at which aTaylor term first produces a larger value than the Taylor term of nextlower degree. That is the t for which

$\begin{matrix}{\frac{t^{n}}{n!} \geq \frac{t^{n - 1}}{( {n - 1} )!}} & (12)\end{matrix}$

This occurs when t≥n. By induction, this tells us that each Taylor termt^(n)/n! will be larger than all other Taylor terms in the intervaln<t<n+1. This trend may be shown in FIG. 24.

This means that as the evaluation interval [−a, a] increases, the termt^(n)/n! with the highest peak will be the one with the largest integern less than a+1. If there were no reference polynomial, therefore, wewould have n_(p)=[a]. An effect of the reference polynomial is to pushthe peak amplitude towards the middle of the evaluation interval.

Since the reference interval applies the same factor to all termst^(n)/n!, it does not affect the relative magnitudes of the terms in anyinterval of t. For instance, if the maximum value produced by any Taylorterm across the evaluation interval is known to have occurred at t=2.5,then the term that produced it can be identified as t²/2!.

In an exemplary embodiment, the combination of the evaluation intervaland the reference polynomial may thus be designed so that the peak valuefor any t^(n)/n! occurs at a t-value that is not close to an integer. Atinteger t-values, there will be two terms t^(n) ^(q) /n_(q)! and t^(n)^(q+1) /n_(q+1)! with the same value; according to an exemplaryembodiment, reducing the PAPR at integer t-values may thus requireremoving two terms, rather than one.

According to an exemplary embodiment, for symmetry, a rough balancebetween the amplitudes of the Taylor terms may be achieved (even beforeRMS normalization). Since the amplitude of the lowest-order term t⁰/0!is equal to 1, in an embodiment, it may thus be desirable for theamplitude of the highest-order term to be roughly 1. For the evaluationinterval [−a, a], if the highest term exponent is x=2^(M)−1, thisimplies that the following should be true:

$\begin{matrix}{\frac{a^{x}}{x!} \cong 1} & (13) \\{a \cong \sqrt[x]{x!}} & (14)\end{matrix}$

Other techniques to control PAPR, including evaluation intervalrestriction and Cairns function removal, may also be understood. In someexemplary embodiments, these techniques may be less effective atcontrolling PAPR than other techniques or may be more situationallyuseful than other techniques, but may be, for example, employed whencontrolling PAPR is less of a concern or in a situation where they areuseful, or may be employed in combination with another technique, asdesired.

For example, according to an exemplary embodiment, evaluation intervalrestriction may be used. In an exemplary embodiment of evaluationinterval restriction, peak size may be controlled within SPDM byshrinking the evaluation interval around t=0, so that the exponentialamplitude change that is characteristic of Cairns functions has littlescope to produce high peaks. However, according to an exemplaryimplementation, this may reduce RMS power as well as peak size. Becauseof this, and because exponential shape is scale-invariant, such anadjustment may not change PAPR. According to some exemplary embodiments,whatever the evaluation interval, SPDM may be capable of generating allpolynomials up to degree 2^(M)−1 within that interval, including oneswith bad PAPR properties.

So, in one exemplary embodiment, the evaluation interval may beshortened, causing the RMS power to be reduced, which may reduce thesignal-to-noise ratio and therefore may increase transmission errors. Inanother exemplary embodiment, the shortened interval may bere-normalized to increase RMS power, which may cause the reappearance ofthe peaks. These adjustments may be made in combination with otheradjustments, if desired.

According to another exemplary embodiment, Cairns function removal maybe used. In an exemplary embodiment, PAPR may be limited by limiting thepolynomial space of SPDM to polynomials exhibiting good PAPR properties,for example by limiting the domain of allowable projection coefficientsfor the Cairns functions. This may entail, for example, determiningwhich Cairns function has the worst PAPR performance over the evaluationinterval (after convolving with the reference polynomial) and removingthat Cairns function from the sub-channel space in order to reduceover-all PAPR.

However, in some exemplary embodiments, this approach may be difficultto implement. The Cairns functions may belong to families with distinctbut related properties. For instance, all Cairns functions ψ_(m,n)(t)with even n are symmetric around zero, and with odd n are anti-symmetricaround zero. This means that sets of Cairns functions may have peaksnear the same points, and in some cases it may be difficult to excludethese function sets.

Instantaneous Spectral Analysis Overview

According to some exemplary embodiments, traditional signal modulationtechniques such as QAM and PSK may transmit signals which are the sum ofsinusoidals of constant amplitude over a symbol period. Bandwidth usagecan be measured within this approach using a Fourier transform, whichrepresents a time domain amplitude sequence in terms of sinusoidals withconstant amplitude.

However, according to an exemplary embodiment, if this approach wereused for SPDM, the result would be very high bandwidth usage. Arbitrarypolynomials do not have concise bandwidth representations in terms ofsinusoidals with constant amplitude.

For instance, according to an exemplary embodiment, if the SPDM basispolynomials are of degree 15, then in 1 TTI they transmit 16 independentamplitude values. If the TTI duration is 1 microsecond, then, accordingto the sampling theorem, at least 8 MHz would be necessary to representthe transmission polynomial. In practice, according to some exemplaryembodiments, a Fourier transform of an SPDM transmission polynomial maygenerally imply much greater spectral occupation than this.

However, Shannon's proof of the sampling theorem uses a Fouriertransform (FT). This proof thereby implicitly assumes the spectrum to bestationary over the evaluation interval (i.e., representable bysinusoidals with constant amplitude).

In an exemplary embodiment, the instantaneous spectral analysis (ISA)technique may allow for transmission of SPDM polynomials usingsinusoidals with continuously-varying amplitude. This means that,regardless of the degree of the transmission polynomial, the range offrequencies B necessary to transmit a sufficient number of points touniquely specify the polynomial can always be made equal to 1/T, where Tis the TTI duration. For instance, for the example given above in whichat minimum 8 MHz are necessary to transmit a polynomial of degree 15using sinusoidals with constant amplitude according to an FT, within ISAonly 1 MHz is necessary.

According to an exemplary embodiment, when additional detail is added tothe time domain amplitude sequence (represented here by increasingpolynomial degree) the FT responses by using higher frequencies to“paint in” the additional detail. However, according to an alternativeexemplary embodiment, the ISA may instead respond by adding additionalsinusoidals with continuously-varying amplitude within the same B=1/Tfrequency range.

Some initial discussion of the differences between the FT and the ISAmay be warranted. The FT and the ISA can be viewed as two differentbasis sets for representing a time-domain amplitude sequence. Whichgives the “correct” representation of the occupied bandwidth depends howthe amplitude sequence is actually transmitted. If the amplitudesequence is transmitted using sinusoidals with constant amplitude, thenthe FT correctly reports where power is placed in the spectrum. If theamplitude sequence is instead transmitted using the ISA-generatedspecification for the amplitude sequence, then ISA correctly reportswhere power is placed in the spectrum.

In an exemplary embodiment, a definition of bandwidth usage that isindependent of choice of basis (in particular, those of the FT or ISA)may be formulated. The economic value of spectrum is tied to how closelychannels can be packed together before they interfere with each other.This suggests that bandwidth occupation should ultimately be measured interms of a channel interference test.

Instantaneous Spectral Analysis

According to an exemplary embodiment, ISA may be used to convert asequence of amplitude values (the “time domain”) into an equivalent sumof sinusoidals with continuously-varying amplitude. By contrast, FTconverts the time domain into a sum of sinusoidals with constantamplitude.

According to an exemplary embodiment, for a particular amplitudesequence, B_(F) may be defined to be the range of sinusoidal frequenciesoccupied by the FT, and B_(I) may be defined to be the range ofsinusoidals with non-zero power as determined by ISA.

Some further discussion of the key differences between ISA and the FTmay be provided.

First, since the FT represents an amplitude sequence with a basis set ofsinusoidals having constant amplitude, it assumes an evaluation periodover which the spectrum is stationary; that is, over which the powerassigned to particular frequencies is constant. This assumes that thesource of the amplitude sequence is Linear Time-Invariant (LTI). ISAdoes not require an LTI source.

Second, the FT effectively averages spectral information over itsevaluation period to produce constant sinusoidal amplitudes. In anexemplary embodiment, ISA may be capable of determiningcontinuously-varying sinusoidal amplitudes at every instant in time(hence the name, “Instantaneous Spectral Analysis”).

Third, for the FT, the maximum rate at which independent amplitudevalues can be transmitted may be equal to the Nyquist rate off_(N)=2B_(F). For the ISA, in an exemplary embodiment, there may be noinherent upper bound in terms of B_(I) on the rate at which independentamplitude values can be transmitted. ISA holds this advantage over theFT because Shannon's proof of the sampling theorem, from which theNyquist rate derives, assumes that the spectrum is stationary over theevaluation interval, which is an assumption that ISA violates.

Stating the above point in a different way, using ISA, it may bepossible to transmit a sequence of amplitude values in a given amount oftime using a smaller range of frequencies than is possible with the FTrepresentation.

To summarize, effectively, for a given amplitude sequence time duration,the FT responds to a higher level of detail in the amplitude sequence byutilizing higher frequencies (increasing B_(F)). ISA responds byincreasing the density of sinusoidals within a constant B_(I).

While the output of the FT may be referred to as the frequency domain,it may alternatively be understood as a frequency domain; it providesone possible representation of the time domain in terms of sinusoidalswith constant amplitude. ISA provides a different frequencyrepresentation in terms of sinusoidals with continuously-varyingamplitude. As both FT and ISA represent occupied bandwidth differently,this raises the question of how occupied bandwidth should be envisionedand represented.

The answer is that it depends on how the time domain was generated. Ifthe source was LTI over the evaluation interval, as with idealtransmitters for traditional modulation techniques that generatesinusoidals with constant amplitude, then the FT representation iscorrect.

However, ISA provides a recipe for generating a non-LTI signal sourcethat will dramatically reduce the necessary range of frequencies fortransmission; for such a source, the FT representation is incorrect.

Practically, bandwidth measurement matters because it affects howclosely channels can be packed together without inter-channelinterference; the differences between FT and ISA may amount todifferences in channel packing.

In an exemplary embodiment, ISA may be implemented based on thefollowing steps. First, a polynomial of interest may be identified. Insome embodiments, this may entail, for example, constructing thepolynomial through amplitude modulation of a polynomial basis set, forexample as previously described. In other embodiments, this may entailfitting a polynomial to an amplitude sequence, such as a time domaininput sequence of real-valued amplitudes. Second, the polynomial may beprojected onto the Cairns series functions. Third, the Cairns seriesfunctions may be converted to the Cairns exponential functions. Last,frequency information contained within the Cairns exponential functionsmay be combined to identify a sum of sinusoidals with continuouslytime-varying amplitudes.

The essential idea behind ISA is to represent the signal time domain asa polynomial, and then to apply a technique to decompose the signalpolynomial into Cairns functions, from which time-varying sinusoidalamplitude information can be determined. The Cairns functions may bedescribed by, for example, Equations (4) and (8) and Table 3, appearingin a previous section.

Referring back to table 3, table 3 shows that the ψ_(m,n)(t)coefficients define a set of orthogonal vectors. More precisely, if M isa positive integer, then the vectors formed from the first 2^(M)coefficients of the functions ψ_(m,n)(t) for 0≤m≤M constitute a set oforthogonal basis vectors for a 2^(M)-dimensional space. These can benormalized to produce the orthonormal “2^(M) Cairns basis vectors”.

The existence of the 2^(M) Cairns basis vectors implies that any Taylorpolynomial P of degree k<2^(M) can be orthogonally projected ontopolynomials formed from the first 2^(M) terms of the Cairns seriesfunctions. This occurs simply by taking the inner product of P'scoefficients with the 2^(M) Cairns basis vectors. The resultingcoefficients for each Cairns basis function are referred to as the“projection coefficients” c_(m,n), also discussed in a previous section.

The first 2^(M) terms of the Cairns series functions ψ_(m,n)(t) are onlyan approximation to the full infinite series expansion of theψ_(m,n)(t). However, the error in the approximation is O(t⁽² ^(M) ⁾),with a reciprocal factorial coefficient, and therefore falls off veryrapidly as M increases. According to an exemplary embodiment,high-degree polynomials may thus be projected onto the ψ_(m,n)(t) bythis procedure with little error.

While spectral usage is not readily apparent from the ψ_(m,n)(t)representation, it can be determined precisely, and on aninstant-by-instant basis, from the equivalent E_(m,n)(t). Essentially,projection onto ψ_(m,n)(t) allows decomposition of a polynomial, andthen representing it as E_(m,n)(t) allows the generation of anequivalent set of sinusoidals with continuously-varying amplitude.

Each E_(m,n)(t) can be expressed as a sum of products, in which eachterm is the product of a phase-adjusted real-valued exponential with acomplex circle. In some embodiments, the real-valued exponentials may beeither rising or decaying, and may have different growth constants inthe exponent. The complex circles may rotate in either direction, andwith different frequencies.

By viewing the real-valued exponentials as continuously time-varyingcoefficients applied to sinusoidals, an approach to defininginstantaneous spectrum can be established. At any particular time, thesum of the real-valued exponentials applied to complex circles of thesame frequency may define the spectral usage at that particularfrequency at that particular time.

In more detail, by using the Euler's formula identity e^(iπ/2)=i it ispossible to represent E_(m,n)(t) as

$\begin{matrix}{{E_{m,n}(t)} = {\frac{1}{\lceil 2^{m - 1} \rceil}{\sum\limits_{p = 0}^{{\lceil 2^{m - 1}\rceil} - 1}{i^{{- {n{({{2p} + 1})}}}2^{2 - m}}e^{\; {t \cdot {\cos {({{\pi {({{2p} + 1})}}2^{1 - m}})}}}}e^{\; {i \cdot t \cdot {\sin {({{\pi {({{2p} + 1})}}2^{1 - m}})}}}}}}}} & (15)\end{matrix}$

In the above equation (15), the phase may be determined by i^(−n(2p+1)2)^(2−m) , the amplitude by e^(t·cos(π(2p+1)2) ^(1−m)) , and the frequencyby e^(i·t·sin(π(2p+1)2) ^(1−m) ⁾.

In an embodiment, the instantaneous spectral information can be found bysumming the phase-weighted amplitude information associated with eachfrequency.

Several points should be noted. First, for m=0 and m=1, the frequencyfactor is equal to the constant 1. Second, for m≥2, no two distinct mlevels may contain the same frequencies, since sin(π(2p+1)2^(1−m))depends on m. Third, the same frequency appears in E_(m,n)(t) for everyn at level m, since sin(π(2p+1)2^(1−m)) does not depend on n. Fourth,since both sin(π(2p+1)2^(1−m)) and cos(π(2p+1)2^(1−m)) can switch signsdepending on the value of p, it follows that for m≥2 each positivefrequency will be matched by an equal negative frequency, and that form≥3 each positive and negative frequency will appear with both a risingand falling exponential as its real-valued amplitude coefficient.

According to an exemplary embodiment, in order to find the instantaneousamplitude of each frequency, all terms that have the same frequency maybe algebraically assembled. Since, as noted above, the frequency factordoes not depend on n, the same frequency information may appear in allE_(m,n)(t) functions having the same m. Phase-adjusted terms may thus besummed across n values at the same m level.

Since a particular frequency is fully-determined by the combination ofits m and p values, a given frequency may be denoted by f_(m,p) and itsamplitude at a particular time by a_(m,p)(t). This may yield thefollowing equation:

$\begin{matrix}{{a_{m,p}(t)} = {\frac{1}{\lceil 2^{m - 1} \rceil}{\sum\limits_{n = 0}^{{\lceil 2^{m - 1}\rceil} - 1}{c_{m,n}i^{{- {n{({{2p} + 1})}}}2^{2 - m}}e^{\; {t \cdot {\cos {({{\pi {({{2p} + 1})}}2^{1 - m}})}}}}}}}} & (16)\end{matrix}$

This equation may provide the instantaneous frequency amplitudesassociated with the polynomial P at every distinct time t over itsevaluation interval. As noted above, for m≥3 each frequency will appeartwice, associated with each of a rising and decaying exponentialamplitude. In an exemplary embodiment, these paired amplitudes may besummed together.

A detail is that the projection coefficients c_(m,n) were found byprojecting Taylor coefficients onto the ψ_(m,n)(t) normalized by theirnumber of non-zero terms, as described above. However, the aboveequation shows the reconstruction of the projected polynomial fromE_(m,n)(t) terms that have not been normalized in this way. SinceE_(m,n)(t)=ψ_(m,n)(t), in an exemplary embodiment, the same projectionnormalization factor as applied to the ψ_(m,n)(t) must be applied formatched m, n values; this was left out for readability.

Instantaneous Spectral Analysis Detailed Operations

Exemplary embodiments of each of the high-level steps provided above maybe provided. For reference, according to an exemplary embodiment, ISAmay be implemented based on the following steps. First, a polynomial maybe fit to the time domain input sequence of real-valued amplitudes.Second, the polynomial may be projected onto the Cairns seriesfunctions. Third, the Cairns series functions may be converted to theCairns exponential functions. Last, frequency information containedwithin the Cairns exponential functions may be combined to identify asum of sinusoidals with continuously time-varying amplitudes.

According to an exemplary embodiment, in a first step, a standardtechnique may be used for fitting a polynomial to a sequence of signalamplitude values. In some embodiments, ISA does not depend on whichmethod is used, and any of several methods may be used. Certainconsiderations may apply, such as whether to produce a high-orderpolynomial that exactly fits the data sequence, or a lower-orderpolynomial that is an adequate fit but perhaps better captures theunderlying pattern. However, in each case, a polynomial may be providedthat represents the time domain amplitude sequence.

For illustrative purposes, FIG. 25 provides MATLAB® code to generate apolynomial of arbitrarily chosen 25th degree, with random Taylorcoefficients between −10 and 10.

According to an exemplary embodiment, in a second step, a polynomial Prepresenting the signal time-domain data may be converted into aweighted sum of Cairns series functions.

First, P may be converted into a Taylor polynomial by multiplyingthrough by the factorial of each term's power. In the MATLAB code sampleshown in FIG. 26, a polynomial is represented as a row vector ofcoefficients called poly_coefficients, with the highest power on theleft. The resulting Taylor_coefficients represents the same polynomialwith the factorials implicit.

According to an exemplary embodiment, code such as that shown in FIG. 26may be capable of handling a matrix in which each row represents adistinct polynomial. However, according to another exemplary embodiment,it may be sufficient to handle matrices in which a single row ispresent.

In the embodiment shown in FIG. 26, to shorten the notation,“Taylor_coefficients” may be referred to equivalently as “Taylor_vec”below.

Next, according to an exemplary embodiment, if necessary or desired,“Taylor_vec” may be padded with minimum-value high-term coefficients sothat the number of coefficients is a power of two, i.e. 2^(M) forpositive integer M. This may be shown in exemplary FIG. 27.

In an exemplary embodiment, a Cairns projection matrix may then beconstructed. FIG. 28 shows an exemplary embodiment of MATLAB code thatmay be used to find the normalization coefficient for each row of theCairns projection matrix. FIG. 29 then shows an exemplary embodiment ofMATLAB code that may be used to, given the normalization coefficients,generate the Cairns projection matrix.

Given the Cairns projection matrix, a Taylor polynomial can be projectedonto Cairns space, for example by simple matrix multiplication. FIG. 30shows an exemplary embodiment of code that may be used to produce a rowvector in which each row is the coefficient for a Cairns seriesfunction.

According to an exemplary embodiment, in a third step, Cairns seriesfunctions may be converted to Cairns exponential functions. Because ofthe identity ψ_(m,n)(t)=E_(m,n)(t), the conversion from Cairns seriesfunctions to Cairns exponential functions may be automatic. Theprojection coefficient for each ψ_(m,n)(t) may simply be applied to thecorresponding E_(m,n)(t).

However, because each ψ_(m,n)(t) was normalized as described above, inan embodiment, the same normalization factors must be applied to theE_(m,n)(t). FIG. 31 shows an exemplary embodiment of MATLAB codeproviding this conversion, in which these normalization coefficients areheld in the proj_norm row vector.

According to an exemplary embodiment, in a fourth step, frequencyinformation may be combined. FIG. 31 shows an exemplary embodiment ofMATLAB code combining amplitude information associated with eachfrequency by summing across E_(m,n)(t) n-values. Note that theamplitudes are time dependent (specified by the evaluation time t).

In the exemplary MATLAB code displayed in FIG. 31, the functionmn_to_row_index may convert from a pair of m, n values to thecorresponding matrix index. This function may be defined in FIG. 32,which may show an exemplary embodiment of MATLAB code that may be usedto convert from a pair of m, n values to the corresponding matrix index.

In an exemplary embodiment, frequency values may be provided in such away that they are interleaved across m-levels. FIG. 33 displays anexemplary embodiment of MATLAB code that may be used to sortfrequencies.

Next, for each frequency, the rising and decaying exponential amplitudesmay be added. In the process, the frequency vectors may be shortened.FIG. 34 displays an exemplary embodiment of MATLAB code that may be usedto combine amplitude pairs.

In the exemplary embodiment shown in FIG. 34, instantaneous frequencyand amplitude information may be stored in short_req_idx andshort_req_amp, respectively.

In an exemplary embodiment, the instantaneous spectral information foundabove can be used to reconstruct the time domain at a particular timevalue t. FIG. 35 displays an exemplary embodiment of MATLAB code thatmay be used to reconstruct the time domain.

In an exemplary embodiment, the accuracy of the reconstruction mayincrease with the degree of the polynomial representing the signal,since the projection onto Cairns space becomes more precise with longerpolynomials. For instance, for a random 25^(th) degree polynomialmaximum percentage reconstruction ratio errors are less than 10⁻¹⁰.

FIG. 36 shows an exemplary plot of the time domain for a random 25^(th)degree Taylor polynomial in the left panel 3602, and its instantaneousspectrum (calculated using the technique given here) in the right panel3604, at the arbitrarily-chosen time value of t=0.28.

In FIG. 36, the curve 3606 in the left panel 3602 shows a random Taylorpolynomial of degree 25. The blue dot 3608 is the ISA reconstruction ofthe polynomial at t=0.28, with the right panel 3604 showing theinstantaneous sinusoidal amplitudes at that time.

According to the exemplary embodiment shown in FIG. 36, the maximumratio percentage error in the ISA reconstruction of the polynomialacross the entire evaluation interval may be 1.2*10⁻¹¹, and the meanreconstruction error may be 4.6*10⁻¹³.

It may be noted from FIG. 36 that, as shown in the left panel 3602, thesimulated transmission time duration T is 1 microsecond. It may also benoted from FIG. 36 that, as shown in the right panel 3604, the range offrequencies B in which ISA puts power is exactly 1 MHz. Since a random25^(th) degree polynomial transmits 26 independent amplitude values, thenumber of amplitude values that can be transmitted in this way is 26BT.This is 13 times higher than the 2BT limit provided by the samplingtheorem on the assumption (which ISA breaks) that the spectrum isstationary over the transmission interval.

This difference is apparent from comparing the mean ISA spectral powerwith the FT of the amplitude sequence generated by the random 25^(th)degree polynomial. FIG. 37 displays an exemplary comparison between themean ISA spectral power with the FT of the amplitude sequence generatedby the random 25^(th) degree polynomial. In the exemplary embodimentshown in FIG. 37, as shown in the left panel 3702, all ISA power isplaced within the 1 MHz range. The FT in the right panel 3704 is on afrequency larger scale, showing only a roughly 40 dB roll-off at 10 MHz.

The foregoing description and accompanying drawings illustrate theprinciples, preferred embodiments and modes of operation of theinvention. However, the invention should not be construed as beinglimited to the particular embodiments discussed above. Additionalvariations of the embodiments discussed above will be appreciated bythose skilled in the art.

Therefore, the above-described embodiments should be regarded asillustrative rather than restrictive. Accordingly, it should beappreciated that variations to those embodiments can be made by thoseskilled in the art without departing from the scope of the invention asdefined by the following claims.

What is claimed is:
 1. A method comprising: converting, with atransmitter, a sequence of amplitude values into a set of sinusoidalswith time-varying amplitude, each of the sinusoidals having a differentfrequency; combining, with the transmitter, the set of sinusoidals withtime-varying amplitude into a signal; transmitting, with thetransmitter, the signal. receiving, with a receiver, the signal.
 2. Themethod of claim 2 where, converting includes converting the sequence ofamplitude values to a Taylor polynomial; projecting the Taylorpolynomial onto a Cairns series function; and converting the projectiononto the Cairns series function to a function described by Cairnsexponential functions to produce the set of sinusoidals withcontinuously time-varying amplitude.